Algebraic Numbers with Elements of Small Height Haydar Göral

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Haydar Göral. Algebraic Numbers with Elements of Small Height. 2017. hal-01191860v3

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ALGEBRAIC NUMBERS WITH ELEMENTS OF SMALL HEIGHT

HAYDAR GORAL¨

Abstract. In this paper, we study the ﬁeld of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer’s conjecture and obtain a model-theoretic characterization of Lehmer’s conjecture for Salem numbers.

1. Introduction The logarithmic height function measures the arithmetic complexity of an algebraic number and it plays a fundamental role in algebra and diophantine geometry. Before defining the logarithmic height function, we first define the Mahler measure of a polynomial over C. For a non-zero polynomial f(x) = ad(X − α1) ··· (X − αd) ∈ C[X], its Mahler measure is defined as the finite product Y m(f) = |ad| |αj|.

|αj |≥1 By convention, the Mahler measure of zero is defined to be 1. Let Q be the field of algebraic numbers. For a non-zero α in Q with irreducible polynomial f(x) ∈ Z[X], we define its Mahler measure as m(α) = m(f). The absolute non-logarithmic height of α is defined by H(α) = m(α)1/d where d = deg f. Then the logarithmic height of α is defined as log m(α) h(α) = log H(α) = . d A central question in diophantine geometry is Lehmer’s conjecture, which is still open and it states that there exists an absolute constant c > 1 such that if m(α) > 1 then m(α) ≥ c. This question was posed by D. Lehmer [14] around 1933, who claimed that the polynomial X10 + X9 − X7 − X6 − X5 − X4 − X3 + X + 1 has the smallest Mahler measure, which is approximately 1.17628 and this is still the smallest known Mahler measure of an algebraic number. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. This is a special case of Lehmer’s conjecture since for a Salem number α we have m(α) = α, and Lehmer’s conjecture for Salem numbers states that 1 is not a limit point of Salem numbers. For a detailed exposition on Mahler measure and Lehmer’s problem, see [20].

Classification theory is an extensive project in model theory, emerged in the work of M. Morley in the 1960’s and S. Shelah in the 1970’s, pursuing the categorization of first-order theories. The question of how many models a theory can have has been at the center of model 1991 Mathematics Subject Classification. 12L12, 11G50, 03C45. Key words and phrases. height, model theory, salem numbers. Partially supported by ValCoMo (ANR-13-BS01-0006) and MALOA (PITN-GA-2009-238381). 1 2 HAYDAR GORAL¨ theory. Stable theories are important to classify their models. If a theory is not stable then its models are difficult to handle and various to classify, and arithmetic fits in this case. A theory T is said to be stable if there is no first-order formula in T which has the order property. Stable theories are equipped with a notion of independence, namely the forking independence, with a well-defined set of properties, reminiscent of the algebraic independence in fields and the linear independence in vector spaces. For instance, the theory of algebraically closed fields is stable. Conversely, stability also yields some structural properties of the models and a well- known result is that a field is ω-stable if and only if it is algebraically closed [15]. A theory is simple, if the forking independence is symmetric and it is known that stable theories are simple. There are many fruitful interactions between model theory and algebra. The main examples are Ax–Grothendieck theorem via compactness [1] and Chevalley’s constructibility theorem via quantifier elimination. More recently, other two important examples are the group configura- tion theorem by Hrushovski [12], which is a generalization of Weil’s group chunk theorem, and Hrushovski’s proof of the Mordell-Lang conjecture for function fields in all characteristics [13]. In this paper, we also study the interaction between model theory and diophantine geometry via the set of algebraic numbers with elements of small height. The results in this article are from the PhD thesis of the author [8].

The model theoretic properties of Q are well understood: it is strongly minimal and has quantifier elimination in the language of rings Lr = {+, −, ·, 0, 1}. In this paper, we mainly focus on the pair Q together with a predicate, in particular our predicate will be the elements of height less than a given positive real number. Moreover, we study these pairs in terms of stability theory. The first motivation came from B. Zilber. In 1990, B. Zilber [24] showed that the pair (C, µ) ≡ (Q, µ) is ω-stable where µ is the group of complex roots of unity, and his pattern was based on the result of H. Mann [16]. The result of B. Zilber [24] was generalized by L. van den Dries and A. Günaydın [6] to algebraically closed fields with a multiplicative subgroup satisfying a certain diophantine condition. In this paper our predicate, namely the elements of small height, contains the group µ. Moreover, the theory of Q with this predicate will be a proper expansion of the theory of (Q, µ). The model theory of pairs has attracted a lot of attention recently. More generally, stable theories with a predicate were analyzed in the paper of E. Casanovas and M. Ziegler [3], where they gave criteria for a pair to be stable. Their result in [3] implies the result of B. Zilber [24] and also B. Poizat’s result on the ω-stability of the theory of pairs of algebraically closed fields [18].

Now we state Kronecker’s theorem from diophantine geometry with no proof, as the details can be found in the book [2, 1.5.9].

Theorem 1.1. (Kronecker) Let α in Q be a non-zero algebraic number. Then h(α) = 0 if and only if it is a root of unity. Combining Zilber’s result [24] with Kronecker’s theorem, we see that the pair (Q, {a ∈ Q : h(a) = 0}) is ω-stable.

Throughout this paper, the language Lm will denote the language {1, ·} where the binary operation · is the usual multiplication. Let S be the set of all Salem numbers. We put × Pb := {a ∈ Q : m(a) ≤ b} and Sb = Pb ∩ S where b ≥ 1, and we set Sε = {a ∈ Q : h(a) ≤ ε} where ε > 0. Both Pb and Sε contain the group of roots of unity µ. In other words, Lehmer’s conjecture and its version for Salem ELEMENTS OF SMALL HEIGHT 3 numbers state that there exists b > 1 such that Pb = P1 = µ and Sb = S1 = ∅ respectively. The pairs (Q, Sb) and (Q,Sε) can be seen as an Lr(U) = Lr ∪ {U} structures where U is a unary relation symbol whose interpretations are Sb and Sε respectively. We call Sε the set of algebraic numbers of small height. Note that the Mahler measure of an algebraic number is again an algebraic number. Put also S = {a ∈ Q : h(a) < 1} = S1, as there is no algebraic number whose height is 1 by Lindemann’s theorem.

Let Tε be the theory T h(Q,Sε). We focus on the model theory of (Q,Sε) in the language Lm(U) = Lm ∪{U} and we prove a result which shows that small perturbations of the property of being a root of unity changes drastically the stability properties of the ambient structure. This means that Tε is model-theoretically very complicated. We also relate the simplicity of a certain pair with Lehmer’s conjecture and give a model-theoretic characterization of a special case of Lehmer’s conjecture. We prove the following theorem (see Theorem (3.4) and Theorem (4.2)).

Main Theorem. The theory of (Q,Sε) is not simple and has the independence property in the language Lm(U). Moreover, the theory of (Q, Sb) is simple for some b > 1 in Lm(U) if and only if Lehmer’s conjecture holds for all Salem numbers. Furthermore, the simplicity of the pair (Q,Pb) for some b > 1 in Lm(U) implies that Lehmer’s conjecture holds for all Salem numbers.

2. Preliminaries and Height Lemmas 2.1. The Logarithmic Height Function. Details of this subsection can be found in [2, Chap- ter 1] and [11, Part B, B.7].

The logarithmic height function behaves well under arithmetic operations and it satisﬁes the following properties: • h(0) = h(1) = 0, • For a non-zero rational number a/b where a and b are coprime, h(a/b) = max{log |a|, log |b|}, • For all α in Q, we have h(αn) = nh(α) where n ∈ N, • For all α and β in Q, we have h(α + β) ≤ h(α) + h(β) + log 2, • For all α and β in Q, we have h(αβ) ≤ h(α) + h(β), • For all non-zero α in Q, we have h(1/α) = h(α). 2.2. Nonstandard Analysis. Next we deﬁne nonstandard extensions which will be used later on.

Definition 2.1. (Nonstandard Extension of a Structure) Let M be a nonempty structure in a countable language L.A nonstandard extension ∗M of M is an ultrapower of M with respect to a non-principal ultrafilter on N. Now let ∗M be a nonstandard extension of M with respect to a non-principal ultrafilter D ∗ on N. Note that the elements of M are of the form (xn)n/D where (xn)n is a sequence in M. ∗ We identify each element x of M with the class of the constant sequence (x)n/D of M. In this way, the structure M becomes an elementary substructure of ∗M. Ultraproduct of structures ∗ automatically become ℵ1-saturated. For a subset A of M, the set A is defined to be the set

{(an)n/D : {n : an ∈ A} ∈ D}. Observe that ∗A contains A as mentioned above. The following sets ∗N, ∗Z, ∗Q, ∗R are called hypernatural numbers, hyperintegers, hyperrational numbers and hyperreals respectively. The elements ∗R \ R are called nonstandard real numbers. Let ∗ Rfin = {x ∈ R : |x| < n for some n ∈ N} 4 HAYDAR GORAL¨

∗ be the ﬁnite numbers. The elements in R \ Rfin are called inﬁnite. Note that the notion of a nonstandard extension and its properties can be extended to many-sorted structures. For more on nonstandard analysis, we direct the reader to [7, 10]. 2.3. Height Lemmas. In this subsection, we compute heights of certain algebraic numbers which will be crucial in the proof of the Main Theorem. Lemma 2.2. For any non-zero algebraic number α which is not a root of unity, the set A(α) = {h(αq): q ∈ Q} is dense in the positive real numbers. Proof. Let α be a non-zero algebraic number which is not a root of unity. Then by Theorem (1.1), we know that h(α) > 0. Furthermore, by the properties of the logarithmic height function given in Section 2.1, we have that h(αq) = |q|h(α) for any q ∈ Q. Given an interval (a, b) where 0 ≤ a < b; choose r ∈ Q such that a/h(α) < r < b/h(α). Then we conclude that r a < h(α ) < b.

Lemma 2.3. Let p1, ..., pk and m1, ..., mk be positive natural numbers. Then

1/m1 1/mk 1/m1 1/mk h(p1 ··· pk ) = h(p1 ) + ··· + h(p1 ).

Proof. Put m = m1 ··· mk and ni = m/mi. As before, we know that for any algebraic number α and rational number q we have that h(αq) = |q|h(α). Therefore, we see that

n1 nk h(p1 ··· pk ) h(p 1/m1 ··· p 1/mk ) = . 1 k m Since for a natural number n ≥ 1 we have h(n) = log n, we conclude that

n1 nk n1 nk h(p1 ··· pk ) h(p1 ) + ··· + h(pk ) = = h(p 1/m1 ) + ··· + h(p 1/mk ). m m 1 1

Lemma 2.4. Let p1, ..., pk, pk+1 and m1, ..., mk, mk+1 be natural numbers such that the product p1 ··· pk and pk+1 are coprime. Then we have p 1/m1 ··· p 1/mk 1 k 1/m1 1/mk 1/mk+1 h 1/m = max{h(p1 ··· pk ), h(pk+1 )}. pk+1 k+1

Proof. Put m = m1 ··· mkmk+1 and ni = m/mi. Recall that for coprime integers a and b, we have h(a/b) = max{log |a|, log |b|} = max{h(a), h(b)}. Thus by the properties of the logarithmic height function in Section 2.1, we have

1/m1 1/mk n1 nk p1 ··· pk 1 p1 ··· pk h 1/m = h n pk+1 k+1 m pk+1 k+1 1 = max{h(p n1 ··· p nk ), h(p nk+1 )}, m 1 k k+1 and the lemma follows from the properties of the logarithmic height function again. Recall that µ is the set of complex roots of unity. Utilizing Kronecker’s theorem, we can deﬁne µ uniformly in the pair (Q,Sε) for any positive ε.

Lemma 2.5. Let φ(x) be the formula x 6= 0 ∧ ∀y(U(y) → U(xy)) in the language Lm(U). Then for any positive ε, the formula φ(x) defines µ in the pair (Q,Sε). In particular, the theory of (Q,Sε) expands the theory of (Q, µ). Proof. Let ε be a positive real number. A root of the unity satisfies the formula since the height function satisfies the inequality h(xy) ≤ h(x) + h(y) and the height of a root of unity is 0. Now suppose α satisfies the formula. If we take y = 1 in the formula, we see that h(α) ≤ ε and so 2 we have α ∈ Sε. Then letting y = α, we get h(α) ≤ ε/2 and thus α ∈ Sε. So taking powers of α, we conclude that h(α) ≤ ε/n for all n ≥ 1 and hence h(α) = 0. The lemma follows from Kronecker’s theorem (1.1). ELEMENTS OF SMALL HEIGHT 5

3. Elements of Small Height In this section, we turn our attention to the model-theoretic properties of the field of algebraic numbers expanded by a predicate to denote elements of small height. We have all the tools to work on these pairs. 3.1. Simplicity and Independence Property. In this subsection we give two definitions for theories, namely simple theories and NIP, which play a significant role in the classification theory introduced by Shelah [19]. Then we prove the first part of the Main Theorem. Definition 3.1. A formula φ(x, y) has the tree property (TP) if there is a tree of parameters <ω (as : ∅= 6 s ∈ ω ) such that <ω (i) For all s ∈ ω ,(φ(x, asi): i < ω) is 2-inconsistent. ω (ii) For all σ ∈ ω ,(φ(x, as): ∅= 6 s ⊂ σ) is consistent. We say that T is simple if no formula has the tree property. Example 3.2. The theory of dense linear orders without end points (DLO) is not simple because the formula ϕ(x, y, z): y < x < z has tree property. Let I be the set of rational numbers <ω in the interval (0, 1) which is a model of DLO. We fix a tree of parameters (qs : ∅ 6= s ∈ ω ) in this model for the formula ϕ above which we will need in order to prove the first part of our Main Theorem. For more on simplicity we refer the reader to [23, Chapter 7]. Definition 3.3. Let T be a complete L-theory where L is a language. An L-formula φ(x, y) is said to have the independence property (IP ) if in every model M of T there is for each n a n family of tuples b1, ..., bn, such that for each of the 2 subsets I of {1, ..., n} there is a tuple aI in M for which

M |= φ(aI , bi) ⇐⇒ i ∈ I. A theory T is called NIP if no formula has IP. In other words, a theory T has the independence property, if there is a ﬁrst-order formula in some model of T that can code any given subset of an arbitrarily large ﬁnite set. It is known that stable theories are NIP. Moreover T is stable if and only if T is simple and NIP. For more on the subject; see [19].

Now we are ready to prove the ﬁrst part of our main theorem using lemmas (2.2), (2.3) and (2.4). Note that Sε is not a group under multiplication (or addition). In lemma (2.5), we proved that the theory of (Q,Sε) expands the theory of (Q, µ) which is ω-stable. On the contrary, the theory of (Q,Sε) is very complicated.

Theorem 3.4. The theory of (Q,Sε) is not simple and has the independence property in the language Lm(U). Proof. Recall that the Mahler measure of an algebraic number is again an algebraic number. Therefore by Lindemann’s theorem, the logarithmic height of an algebraic number α is transcendental if h(α) 6= 0. First we prove that the theory is not simple. In order to show this, we exhibit a formula which has the tree property. Actually, the argument from Case 2 below also works in Case 1, and Case 1 is included to show the main idea of the proof in a more transparent way.

Case 1: The element ε is in the range of the logarithmic height function, that is to say ε = h(α) for some α ∈ Q. Now put ty tx φ(x, y, z, t): U ∧ U . x z 6 HAYDAR GORAL¨

Observe that for any rational number r, s ∈ (0, 1), αr (1) h α ≤ ε ⇐⇒ r ≤ s. αs

Furthermore if (r1, s1) and (r2, s2) are disjoint intervals of (0, 1), where r1 < s1 < r2 < s2 ∈ Q, then we cannot have

r1 s1 r2 s2 (Q,Sε) |= ∃x(φ(x, α , α , α) ∧ φ(x, α , α , α)), otherwise we have x αr2 h α ≤ ε and h α ≤ ε. αs1 x 2 αr2 Therefore, if we multiply these elements, we obtain that h α αs1 ≤ 2ε. This is a contradiction,

2 m β β1 βm since s1 < r2 and h(α ) = 2ε. For a tuple β = (β1, ..., βm) ∈ Q , we put α = (α , ..., α ). Combining (1) and the previous argument, we see that the formula φ(x, y, z, t) has the tree qs <ω property if we take the parameters (α , α) where the parameters (qs : ∅ 6= s ∈ ω ) are as given in example (3.2).

Case 2: The element ε is not in the range of the logarithmic height function. Without loss of generality, we can assume that ε = 1 i.e Sε = S as 1 is not transcendental. This time we use the fact that the range of the logarithmic height function is dense in the positive reals. Again we set ty tx φ(x, y, z, t): U ∧ U . x z We will show that this formula has the tree property by finding some parameters in some model ∗ of Tε. Let M be a nonstandard extension of the many-sorted structure x M = (Q, +, −, ·, 0, 1, h, R≥0, 2 , <, Q). Then the logarithmic height function extends to ∗Q and it takes values in positive hyperreal numbers. We also denote this extension as h. Then the pair (∗Q, ∗S) is an elementary extension ∗ of (Q,S) in Lm(U). Note that S is the set of hyperalgebraic numbers whose heights are less than 1. Let st(a) denote the standard part of a finite hyperreal number. By lemma (2.2), we know that the sequence {h(2k)} is dense in positive real numbers where k is a rational number. In particular, there is a hyperrational number q such that q < 1/ log 2 and st(h(2q)) = 1. In other words, q < 1/ log 2 and it is infinitely close to the real number 1/ log 2. Observe again that for any rational number r, s ∈ (0, 1), 2r h 2q ≤ 1 ⇐⇒ r ≤ s. 2s

Furthermore if (r1, s1) and (r2, s2) are disjoint intervals of (0, 1), where r1 < s1 < r2 < s2 ∈ Q, then we cannot have ∗ ∗ r s q r s q ( Q, S) |= ∃x(φ(x, 2 1 , 2 1 , 2 ) ∧ φ(x, 2 2 , 2 2 , 2 )), otherwise we have x 2r2 h 2q ≤ 1 and h 2q ≤ 1. 2s1 x Therefore, if we multiply these elements, we obtain that 2r2 h 22q ≤ 2. 2s1 2q Similarly this is a contradiction, as s1 < r2 and st(h(2 ) = 2 imply 2r2 h 22q > 2. 2s1 ELEMENTS OF SMALL HEIGHT 7

Thus the formula φ(x, y, z, t) has the tree property if we take the parameters (2qs , 2q) where <ω qs the parameters (qs : ∅= 6 s ∈ ω ) are as given in example (3.2), and the tuple 2 is deﬁned as in Case 1. Hence Tε is not simple.

Now we show that Tε has the independence property. For simplicity, we assume that ε = 1. Let φ(x, y) be the formula U(x/y).

We will show that this formula has IP. Let n ≥ 1 be given. Let p1, ..., pn be distinct prime km numbers. Put bm = pm for m = 1, ..., n, where km ∈ Q will be chosen later. Now let I be a subset of {1, ..., n} and r be the size of I. For i ∈ I, by lemma (2.2) we can choose ei ∈ Q such that 1 1 1 < h(p ei ) ≤ + r i r (n + 1)2 and we set

Y ei aI = pi . i∈I By lemma (2.3), we have 1 1 < h(a ) ≤ 1 + . I n + 1

Now by lemma (2.2), choose km such that 1 < h(b ) < 1. n + 1 m

Finally by lemma (2.4) and the properties of the height function, we get φ(aI , bi) if and only if i ∈ I. This yields that φ(x, y) has the independence property and hence Tε is not NIP.

4. Salem Numbers and Lehmer’s conjecture Lehmer’s conjecture (1933): There exists an absolute constant c > 1 such that if α is not a root of unity nor zero, then m(α) ≥ c.

This conjecture stands still open. The best known result is due to E. Dobrowolski [5] in 1979, who proved that for d ≥ 2, 1 (log log d)3 m(α) > 1 + , 1200 log d where d = deg(α). A polynomial p(x) of degree d is called reciprocal if xdp(1/x) = ±p(x). Furthermore, an algebraic number α is reciprocal if it is Galois conjugate to 1/α, in other words its minimal polynomial is reciprocal. In 1971, C. Smyth [22] showed that for α a nonreciprocal algebraic number m(α) ≥ m(x3 − x − 1) = θ ≈ 1.3247, where θ is the real root of x3 − x − 1. Thus the remaining and the difficult case for Lehmer’s conjecture is the set of reciprocal algebraic numbers. The next definition plays a central role in Lehmer’s conjecture. Definition 4.1. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all others Galois conjugates of α lie on the unit circle. So a Salem number is reciprocal. Observe that for a Salem number α and a positive integer n, the element αn is also a Salem number, and we have m(αn) = m(α)n = αn. 8 HAYDAR GORAL¨

In general this is not true for an arbitrary algebraic number. It is an open question whether 1 is a limit point of Salem numbers. This is a special case of Lehmer’s conjecture. The smallest known Salem number α was given by D. Lehmer [14] which is the root of the polynomial X10 + X9 − X7 − X6 − X5 − X4 − X3 + X + 1 with α ≈ 1.17628 and not many Salem numbers are known in the interval (1, 1.3). The list of small Salem numbers can be found in the link [25]. For more about Lehmer’s conjecture and Salem numbers, we direct the reader to [20, 21]. The rest of the paper is devoted to Lehmer’s conjecture via model theory. Let S be the set of all Salem numbers. Recall that × Pb := {a ∈ Q : m(a) ≤ b} and Sb = Pb ∩ S where b ≥ 1. Note that Sb is the set of Salem numbers ≤ b. Lehmer’s conjecture for Salem numbers is equivalent of there exists b > 1 such that Sb = S1 = ∅. The next Theorem relates the simplicity of the pair (Q, Sb) with Lehmer’s conjecture and it is the other part of the Main Theorem. In particular, it states that the simplicity of the pair (Q, Sb) for some b > 1, which is a weaker statement than ω-stability, is equivalent of Lehmer’s conjecture for Salem numbers.

Theorem 4.2. The theory of (Q, Sb) is simple for some b > 1 in Lm(U) if and only if Lehmer’s conjecture holds for all Salem numbers. Moreover, the simplicity of the pair (Q,Pb) for some b > 1 in Lm(U) implies that Lehmer’s conjecture holds for all Salem numbers.

Proof. If Lehmer’s conjecture is true for Salem numbers, then we know that the pair (Q, Sb) = (Q, ∅) is ω-stable for some b > 1. Now suppose that (Q, Sb) is simple for some b > 1. Assume that 1 is a limit point of Salem numbers in order to get a contradiction. Then we can choose a Salem number α and a positive integer n such that αn is very close to b. Denote the integer ∗ ∗ ∗ ∗ ∗ part of a as [a]. Let M = ( Q, +, −, ·, 0, 1, m, R≥1, Q, N, <, g, [·]) be a nonstandard extension of the many-sorted structure M = (Q, +, −, ·, 0, 1, m, R≥1, Q, N, <, g, [·]) where g is the binary function from Q × N to Q defined as g(a, b) = ab. The Mahler measure m extends to ∗Q and it takes values in positive hyperreal numbers ≥ 1. We also denote this extension by m. Now ∗ ∗ ∗ the pair ( Q, Sb) is an elementary extension of (Q, Sb) in Lm(U). Note that Sb is the set of hyperalgebraic Salem numbers that are less than b. Then there is a nonstandard Salem number β > 1 which is infinitely close to 1, and an infinite nonstandard natural number N such that βN < b and st(βN ) = b. Put ty tx φ(x, y, z, t): U ∧ U x z as before. Observe that for any rational numbers r and s in (0, 1), we have β[Nr] m βN ≤ b ⇐⇒ r ≤ s β[Ns] since N is infinite. Similar to the proof of Theorem (3.4), the formula φ(x, y, z, t) has the tree [Nqs] N property by taking the parameters as = (β , β ) which are coming from example (3.2). Hence (Q, Sb) is not simple, a contradiction. The proof of the second part follows from the methodology for proving the first part of the theorem. Remark 4.3. We call an algebraic number α multiplicative if m(αn) = m(α)n for all natural numbers n ≥ 1. Let Db be the set of multiplicative algebraic numbers whose Mahler measure is less than b. Note that Db contains Sb. Then the same proof of Theorem (4.2) shows that the simplicity of the pair (Q,Db) for some b > 1 in Lm(U) is equivalent to Lehmer’s conjecture for multiplicative algebraic numbers. The following conjecture is the model-theoretic characterization of Lehmer’s conjecture.

Conjecture 4.4. The theory of (Q,Pb) is stable for some b > 1 in Lm(U). Moreover the stability of the pair (Q,Pb) in Lm(U) for some b > 1 implies Lehmer’s conjecture. ELEMENTS OF SMALL HEIGHT 9

One way of showing the simplicity is to ﬁnd a notion of independence which is symmetric and satisfying the axioms of non-forking [23, Chapter 7]. In [8, 9], the author characterized the independence in a suﬃciently saturated model (U, G) of (Q, µ) in the language Lr(U). More precisely, let A and B be two algebraically closed sets in the pair (U, G). Then

(U,G) ACF A ^| B if and only if A ^| B, G. A∩B A∩B,G

Hence in a suﬃciently saturated model of (Q,Pb) for a small b, if the theory is simple then the independence should be the independence given above.

When 1 < b < 1.3, by [20] we know that every element α in Pb is reciprocal. By [3, 17], every × formula in the pair (Q ,Pb) is bounded, which means every formula in the pair is equivalent to a formula of the form Qα ∈ Pb φ(x, α) where Q is a tuple of quantifiers and φ(x, α) is an Lm-formula. Moreover by [3, 17] again, if the induced structure on Pb is stable or does not have × the finite cover property, then the pair (Q ,Pb) is stable. Hence, understanding the induced structure on Pb and how Pb (a set of reciprocal algebraic integers) intersects a set of the form × a1 an {(x1, ..., xn): x1 ··· xn = 1, xi ∈ Q , ai ∈ Z} will lead to the stability of the pair, thus to × Lehmer’s conjecture. Another direction is first to show a weaker result that the pair (Q ,Pb) is NIP in Lm(U). By the paper of A. Chernikov and P. Simon [4], again we need to understand × the induced structure on Pb. If the induced structure on Pb is NIP, then so is the pair (Q ,Pb). We end the paper by posing the following question which is also related to Lehmer’s conjecture.

• Is µ deﬁnable in (Q,Pb) for some b > 1?

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Department of Mathematics, Koc¸University, Rumelifeneri Yolu, 34450, Sarıyer, Istanbul, TURKEY. E-mail address: [email protected]